BENCHMARK PROBLEMS IN STRUCTURAL …sstl.cee.illinois.edu/benchmarks/bench1journal/BENCH.pdf · relative effectiveness and implementability of various structural control algorithms

Dec. 7, 1997 1 Spencer,

et al.

BENCHMARK PROBLEMS IN STRUCTURAL CONTROL:PART I – ACTIVE MASS DRIVER SYSTEM

B.F. Spencer Jr.,

1

S.J. Dyke

2

and H.S. Deoskar

3

SUMMARY

This paper presents the overview and problem definition for a benchmark structural controlproblem. The structure considered — chosen because of the widespread interest in this class ofsystems (Soong 1990; Housner,

et al.

1994b; Fujino,

et al.

1996) — is a scale model of a three-story building employing an active mass driver. A model for this structural system, including theactuator and sensors, has been developed directly from experimentally obtained data and willform the basis for the benchmark study. Control constraints and evaluation criteria are presentedfor the design problem. A simulation program has been developed and made available to facilitatecomparison of the efficiency and merit of various control strategies. A sample control design isgiven to illustrate some of the design challenges.

INTRODUCTION

Tremendous progress has been made over the last two decades toward making active struc-tural control a viable technology for enhancing structural functionality and safety against naturalhazards such as strong earthquakes and high winds. The success of the First World Conference onStructural Control, held in Pasadena, California in August 1994, demonstrated the world-wide in-terest in structural control (Housner,

et al.

1994a). The Conference attracted over 300 participantsfrom 15 countries. The Second World Conference on Structural Control, to be held in Kyoto, Ja-pan in the summer of 1998, promises to continue in this tradition.

Since the initial conceptual study by Yao (1972), many control algorithms and devices havebeen investigated, each with its own merits, depending on the particular application and desiredeffect. Clearly, the ability to make direct comparisons between systems employing these algo-rithms and devices is necessary to focus future efforts in the most promising directions and to ef-fectively set performance goals and specifications. Indeed, the development of guidelinesgoverning both performance and implementability is a meaningful and important task in itself.

This paper presents a benchmark structural control problem that can be used to evaluate therelative effectiveness and implementability of various structural control algorithms and to providean analytical

testbed

for evaluation of control design issues such as model order reduction, spill-over, control-structure interaction, limited control authority, sensor noise, available measure-ments, computational delay,

etc.

To achieve a high level of realism, an

evaluation

model ispresented in the problem definition which is derived directly from experimental data obtained atthe Structural Dynamics and Control/Earthquake Engineering Laboratory (SDC/EEL) at the Uni-versity of Notre Dame. This model accurately represents the behavior of the laboratory structureand fully incorporates actuator/sensor dynamics. Herein, the evaluation model will be considered

1. Prof., Dept. of Civil Engrg. and Geo. Sci., Univ. of Notre Dame, Notre Dame, IN 46556-0767.2. Assist. Prof., Dept. of Civil Engrg., Washington Univ., St. Louis, MO 63130-4899.3. Former Grad. Assist., Dept. of Civil Engrg. and Geo. Sci., Univ. of Notre Dame, Notre Dame, IN 46556-0767.

Published in

Earthquake Engineering and Structural Dynamics

,

27

(11), 1998, 1127–1139.


et al.

as the real structural system. In general, controllers that are successfully implemented on the eval-uation model can be expected to perform similarly in the laboratory setting. Several evaluationcriteria are given, along with the associated control design constraints. A sample control design ispresented to illustrate some of the design challenges. This benchmark problem can be viewed asan initial step toward development of standardized performance evaluation procedures.

EXPERIMENTAL STRUCTURE

The structure on which theevaluation model is based is an ac-tively controlled, three-story, single-bay, model building considered inDyke,

et al.

(1994a, 1996a). The teststructure, shown in Figs. 1 and 2, isdesigned to be a scale model of theprototype building discussed inChung,

et al

. (1989) and is subject toone-dimensional ground motion. Thebuilding frame is constructed of steel,with a height of 158 cm. The floormasses of the model totaled 227 kg,distributed evenly between the threefloors. The structural frame mass was77 kg. The time scale factor is 0.2,making the natural frequencies of themodel approximately five times thoseof the prototype. The first threemodes of the model structural systemare at 5.81 Hz, 17.68 Hz and 28.53Hz, with associated damping ratiosgiven, respectively, by 0.33%, 0.23%,and 0.30%. The ratio of model quan-tities to those corresponding to theprototype structure are: force = 1:60,mass = 1:206, time = 1:5, displace-ment = 4:29 and acceleration = 7:2.

For control purposes, a simple implementation of an active mass driver (AMD) was placedon the third floor of the structure. The AMD consists of a single hydraulic actuator with steelmasses attached to the ends of the piston rod (see Fig. 3). The servo-actuated hydraulic cylinderhas a 3.8 cm diameter and a 30.5 cm stroke. For this experiment, the moving mass for the AMDwas 5.2 kg, and consisted of the piston, piston rod, and steel disks bolted to the end of the pistonrod. The total mass of the structure, including the frame and the AMD, was 309 kg. Thus, themoving mass of the AMD is 1.7% of the total mass of the structure. Because hydraulic actuatorsare inherently open loop unstable, position feedback was employed to stabilize the control actua-tor. The position of the actuator was obtained using an LVDT (linear variable differential trans-former), rigidly mounted between the end of the piston rod and the third floor.

Figure 1. Three Degree-of-Freedom Test Structure with AMD System.


et al.

Structural displace-ments and velocities aredifficult to obtain directly infull scale structures, be-cause they must be mea-sured relative to an inertialreference frame. Alterna-tively, acceleration mea-surements can readily beacquired at arbitrary loca-tions on the structure. Forthis experiment, accelerom-eters were positioned on theground, on each floor of thestructure, and on the AMD,as shown in Fig. 2. The dis-placement of the AMD rela-tive to the third floor wasalso measured using theLVDT mentioned above.Thus, the measurementsthat are directly availablefor control force determina-tion are the three floor ac-celeration measurements,the ground acceleration,and the displacement andacceleration of the AMD(see Fig. 2). Additionally,pseudo absolute velocitiesare available by passing themeasured accelerations through a second order filter that is essentially a high-pass filter in serieswith an integrator (Ivers and Miller, 1991).

EVALUATION MODEL

A high-fidelity, linear time-invariant state space representation of the input-output model forthe structure described in the previous section has been developed. The model has 28 states and isof the form

(1)

(2)

(3)

xa2

xa3

xa1

xm xam,

xg

DSP Board &

Control ComputerControlActuator

Figure 2. Schematic Diagram of Experimental Setup.

x Ax Bu E xg+ +=

y Cyx Dyu Fyxg v+ + +=

z Czx Dzu Fzxg+ +=


et al.

where is the state vector, is the scalar ground acceleration, is the scalar control input, is the vector of responses that can be directly measured,

is the vector of responses that can be regu-lated. Here, is the displacement of the

i

th floor relative to the ground, is the displacement ofthe AMD relative to the third floor, is the absolute acceleration of the

i

th floor, is the ab-solute acceleration of the AMD mass, is the vector of measurement noises, and

, , and are matrices of appropriate dimension. The coefficient ma-trices in Eqs. (1–3) are determined from the data collected at the SDC/EEL using the identifica-tion methods presented in Dyke,

et al

. (1994a,b, 1996a,b). The resulting model represents theinput-output behavior of the structural system up to 100 Hz and includes the effects of actuator/sensor dynamics and control-structure interaction. Figure 4 provides a representative comparisonbetween the model from Eqs. (1–3) and the experimental data. Note that the experimental datawas obtained using a 16-bit data acquisition board. Thus, the difference between the experimentaldata and the model in Fig. 4a at low frequencies is due to the finite precision of the data acquisi-tion system.

The model given in Eqs. (1–3) is termed the

evaluation

model and will be used to assess theperformance of candidate controllers; that is, the evaluation model is considered herein to be thetrue representation of the structural system.

CONTROL DESIGN PROBLEM

The control design problem is to determine a discrete-time, feedback compensator of theform

(4)

(5)

Figure 3. Active Mass Driver.

x xg uy xm xa1 xa2 xa3 xam xg, ,,, ,[ ]′=z x1 x2 x, , 3 xm x1 x2 x, , 3 xm xa1 xa2 xa3 xam, ,, , , , ,[ ]′=

xi xmxai xam

vA B E Cy Dy Cz Dz, , , , , , Fy Fz

xk 1+c

f 1(xkc y

˜ kuk k, ), ,=

uk f 2(xkc y

˜ kk, ),=

Dec. 7, 1997 5 Spencer, et al.

Figure 4. Representative Comparison of the Transfer Functions for the Test Structure and the Evaluation Model. (a) Actuator Command to 1st Floor Absolute Acceleration, (b) Actuator

Command to AMD Absolute Acceleration.

Frequency (Hz)

Frequency (Hz)

Pha

se (

deg)

Mag

nitu

de (

dB)

(a)

(b)

0 10 20 30 40 50 60 70 80 90 100−200

−100

0

100

200

Experimental DataEvaluation Model

0 10 20 30 40 50 60 70 80 90 100−60

−40

−20

0

20

0 10 20 30 40 50 60 70 80 90 100−200

−100

0

100

200

Experimental DataEvaluation Model

0 10 20 30 40 50 60 70 80 90 100

−100

−50

0

50

Pha

se (

deg)

Mag

nitu

de (

dB)


where , and are the state vector for the compensator, the output vector and the controlcommand, respectively, at time . For this problem the performance of all control designsmust be assessed using the evaluation model described previously. For each proposed control de-sign, performance and stability robustness should be discussed. As detailed in the following para-graphs, the merit of a controller will be based on criteria given in terms of both rms and peakresponse quantities. Normally, smaller values of the evaluation criteria indicate superior perfor-mance.

Evaluation Criteria: RMS Responses

Assume that the input excitation is a stationary random process with a spectral densitydefined by the Kanai-Tajimi spectrum

(6)

where and are unknown, but assumed to lie in the following ranges:, . To have a basis for comparison, the spectral in-

tensity is chosen such that the rms value of the ground motion takes a constant value of g’s, i.e.,

g2.sec (7)

The first criterion on which controllers will be evaluated is based on their ability to mini-mize the maximum rms interstory drift due to all admissible ground motions. Therefore, the non-dimensionalized measure of performance is given by

(8)

where is the stationary rms interstory drift for the ith floor, and cm is the worst-case stationary rms displacement of the third floor of the uncontrolled building over the class ofexcitations considered (occurring when rad/sec, ). The interstory drifts aregiven respectively by , and .

A second evaluation criterion is given in terms of the maximum rms absolute acceleration,yielding a performance measure given by

(9)

where is the stationary rms acceleration for the ith floor, and g’s is the worst-case stationary rms acceleration of the third floor of the uncontrolled building (occurring when

rad/sec, ).

xkc y

˜ kuk

t kT=

xg

Sxgxgω( )

S0(4ζg2ωg

2ω2 ωg4)+

(ω2 ωg2)–2

4ζg2ωg

2ω2+

-------------------------------------------------------=

ωg ζg20 rad/sec ωg 120 rad/sec≤ ≤ 0.3 ζg 0.75≤ ≤

σ xg0.12=

S0

0.03ζg

πωg(4ζg2

1)+---------------------------------=

J1 maxσd1

σx3o

---------σd2

σx3o

---------σd3

σx3o

---------, ,

=ωg ζg,

σdiσx3o

1.31=

ωg 37.3= ζg 0.3=d1 t( ) x1 t( )= d2 t( ) x2 t( ) x1 t( )–= d3 t( ) x3 t( ) x2 t( )–=

J2 maxσ xa1

σ xa3o

-----------σ xa2

σ xa3o

-----------σ xa3

σ xa3o

-----------, ,

=ωg ζg,

σ xaiσ xa3o

1.79=

ωg 37.3= ζg 0.3=


The hard constraints for the control effort are given by volt, g’s and. Additionally, candidate controllers are to be evaluated based on the required control

resources. Three quantities, and , should be examined to make the assessment. Therms actuator displacement, , provides a measure of the required physical size of the device.The rms actuator velocity, , provides a measure of the control power required. The rms abso-lute acceleration provides a measure of the magnitude of the forces that the actuator mustgenerate to execute the commanded control action. Therefore, the nondimensionalized control re-source evaluation criteria are

(10)

(11)

(12)

where cm/sec is the worst-case stationary rms velocity of the third floor relative tothe ground for the uncontrolled structure (occurring when rad/sec, ).

Evaluation Criteria: Peak Responses

Here, the input excitation is assumed to be a historical earthquake record. Both the 1940El Centro NS record and the NS record for the 1968 Hachinohe earthquake should be considered.Because the system under consideration is a scale model, the time scale should be increased by afactor of 5 (i.e., the earthquakes occur in 1/5 the recorded time). The required scaling of the mag-nitude of the ground acceleration is 3.5. The evaluation criterion is based on minimization of thenondimensionalized peak interstory drifts due to both earthquake records. For each earthquake,the maximum drifts are nondimensionalized with respect to the uncontrolled peak third floor dis-placement, denoted , relative to the ground. Therefore, the performance measure is given by

(13)

A second performance evaluation criterion is given in terms of the peak acceleration, yield-ing

(14)

where the accelerations are nondimensionalized by the peak uncontrolled third floor acceleration,denoted , corresponding respectively to each earthquake.

σu 1≤ σ xam2≤

σxm3 cm≤

σxmσxm

, σxam

σxm

σxm

σ xam

J3 maxσxm

σx3o

---------

=ωg ζg,

J4 maxσxm

σx3o

---------

=ωg ζg,

J5 maxσ xam

σ xa3o

-----------

=ωg ζg,

σx3o47.9=

ωg 37.3= ζg 0.3=

xg

x3o

J6 max maxd1 t( )x3o

---------------d2 t( )x3o

---------------d3 t( )x3o

---------------, ,

=tEl Centro

Hachinohe

J7 max maxxa1 t( )xa3o

-----------------xa2 t( )xa3o

-----------------xa3 t( )xa3o

-----------------, ,

=tEl Centro

Hachinohe

xa3o


The control constraints are volts, cm, g’s, and both the El Centro and the Hachinohe earthquakes should again be considered. Addition-ally, the candidate controllers are to be evaluated in terms of the required control resources as fol-lows

(15)

(16)

(17)

where is the peak uncontrolled third floor relative velocity corresponding respectively to eachearthquake.

For the El Centro earthquake, cm, cm/sec and g’s.For the Hachinohe earthquake, cm, cm/sec and g’s.

CONTROL IMPLEMENTATION CONSTRAINTS

To make the benchmark problem as realistic as possible, the following implementation con-straints are placed on the system:

1. As indicated previously, the measurements that are directly available for use in determinationof the control action are . Although absolute velocities are notavailable, they can be closely approximated by passing the measured accelerations through asecond order filter with the following transfer function

(18)

where is the pseudo velocity response in that it will track the absolute velocity responseabove 1 Hz. Therefore, the pseudo velocities, , are also available for deter-mination of the control action, and the combined output vector is given by

. For more information regarding prac-tical issues associated with implementing the filter in Eq. (18), see Ivers and Miller (1991).

2. The controller for the structure is digitally implemented with a sampling time of sec.

3. A computation delay of 200 sec is required to perform the D-matrix calculations in the con-trol action determination and for the associated A/D and D/A conversions (Quast, et al., 1995).

4. The A/D and D/A converters on the digital controller have 12-bit precision and a span of 3 V.

max u t( ) 3≤ t

max xm t( ) 9≤ t

max xam t( ) 6≤ t

J8 max max xm t( )x3o

----------------=tEl Centro

Hachinohe

J9 max max xm t( )x3o

----------------=tEl Centro

Hachinohe

J10 max max xam t( )xa3o

------------------=tEl Centro

Hachinohe

x3o

x3o 3.37= x3o 131= xa3o 5.05=x3o 1.66= x3o 58.3= xa3o 2.58=

y xm xa1 xa2 xa3 x, am xg, , , ,[ ]′=

Hx˜˙x s( ) 39.5s

39.5s2

8.89s 1+ +--------------------------------------------=

x˜˙

x˜˙a1 x

˜˙a2 x

˜˙a3 x

˜˙am x

˜˙g, , , ,

y˜

xm xa1 xa2 xa3 x, am xg x˜˙a1 x

˜˙a2 x

˜˙a3 x

˜˙am x

˜˙g,, , , , , , , ,[ ]′=

T 0.001=

µ

±


5. Each of the measured responses contains an rms noise of 0.01 Volts, which is approximately0.3% of the full span of the A/D converters. The measurement noises are modeled as Gaussianrectangular pulse processes with a pulse width of 0.001 seconds.

6. To account for limited computational resources in the digital controller, the controller given inEqs. (4) and (5) is restricted to have no more than 12 states.

7. The performance of each control design should be evaluated using the 28 state evaluation mod-el given in Eqs. (1)–(3).

8. The controller given in Eqs. (4) and (5) is required to be stable.

The SIMULINK (1994) model shown in Fig. 5 has been developed to simulate the features andlimitations of this structural control problem. Note that, although the controller is digital, thestructure is still modeled as a continuous system. To reduce integration errors, a time step of0.0001 sec is used in the simulation.

SAMPLE CONTROL DESIGN

To illustrate some of the constraints and challenges of this benchmark problem, a samplelinear quadratic Gaussian (LQG) control design is presented. The first step in this process is to de-

Figure 5. SIMULINK Model for the Benchmark Problem.

u

z

y˜

xg


velop a reduced order model, designated the design model, for purposes of control design. The de-sign model has the form

(19)

(20)

where is a 10-dimensional state vector, and , , , , and are the reduced order coefficient matrices. This design will not make use of the measure-ment of the ground excitation, the pseudo absolute velocities or the actuator displacement, al-though these measurements are available.

To simplify design of the controller, is taken to be a stationary white noise, and an infi-nite horizon performance index is chosen that weights the accelerations of the three floors, i.e.,

(21)

where , and all of the elements of the weighting matrix Q are zero, except for. Further, the measurement noise is assumed to be identically distributed,

statistically independent Gaussian white noise processes, and .The separation principle allows the control and estimation problems to be considered sepa-

rately, yielding a controller of the form (Stengel 1986; Skelton 1988)

(22)

where is the Kalman Filter estimate of the state vector based on the reduced order model. Bythe certainty equivalence principle (Stengel 1986; Skelton 1988), is the full state feedback gainmatrix for the deterministic regulator problem given by

(23)

where is the solution of the algebraic Riccati equation given by

(24)

and

(25)

(26)

(27)

(28)

xr A rxr Bru Er xg+ +=

yr Cyrxr Dyru Fyr xg vr+ + +=

xr yr xa1 xa2 xa3 x, am, ,[ ]′= A r Br Er Cyr DyrFyr

xg

J lim 1τ--- E Cyrxr Dyru+( )′Q Cyrxr Dyru+( ) ru

2+{ } td

0

τ

∫=τ ∞ →

r 50=Q11 Q22 Q33 1= = =

SxgxgSvivi

⁄ γ 25= =

u Kx r–=

xrK

K N ′ Br ′P+( ) r⁄=

P

0 PA A ′P PBrBr ′P r⁄ Q+–+=

Q Cyr ′QCyr NN ′ r⁄–=

N Cyr ′QDyr=

r r Dyr ′QDyr+=

A A r Br N ′ r⁄–=


Calculations to determine were done using the MATLAB (1994) routine lqry.m within the con-trol toolbox.

The Kalman Filter optimal estimator is given by

(29)

(30)

where is the solution of the algebraic Riccati equation given by

(31)

and

(32)

(33)

(34)

(35)

Calculations to determine were done using the MATLAB routine lqew.m within the controltoolbox.

Finally, the controller is put in the form of Eqs. (4–5) using the bilinear transformation (An-toniou, 1993) to yield the following compensator

(36)

(37)

As required, the .To assess the performance of the sample controller, it is implemented on the evaluation

model discussed previously. Based on an eigenvalue analysis, both the controller and the closed-loop system are stable. The loop gain transfer function was used to provide an indication of theclosed-loop stability of the system. Here, the loop gain transfer function is defined as the transferfunction of the system formed by breaking the control loop at the input to the system (Dyke, et al.,1994b, 1996a,b). The loop gain formed with this control design is provided in Fig. 6. A controldesign was considered to be robust if the magnitude of the loop gain was below –5 dB at all fre-quencies above 35 Hz.

For the first five evaluation criteria, the rms values of the constraint variables are cm, g and V. For evaluation criteria six through ten, the

peak values of the constraint variables are cm, g and V. Thus,all control design constraints were achieved with this control design. The associated evaluation

K

xr A r xr Bru L y C yr xr– Dyru–( )+ +=

L [R˜

1– γFyrEr ′ CyrS+( )] ′=

S

0 SA˜

A˜

′S SG˜

S H˜

+–+=

A˜

A r ′ Cyr ′R˜1– γFyrEr ′( )–=

G˜

Cyr ′R˜1– Cyr=

H˜

γErEr ′ γ2ErFyr ′R˜1– FyrEr ′–=

R˜

I γFyrFyr ′+=

L

xk 1+c Acxk

c Bcyk+=

uk Ccxkc Dcyk+=

dim(xkc) 10 12≤=

σxm0.671= σ xam

1.12= σu 0.143=xm 2.00= xam 4.83= u 0.526=


criteria are given in Table 1. Note that the rms and peak responses were determined through simu-lation using the SIMULINK program shown in Figure 5. The rms responses were calculated as-suming an ergodic response and averaging over a 300 second time period.

CLOSURE

The 28-state evaluation model, as well as the 10-state control design model, the MATLAB(1994) m-file used to do the sample control design, the input data and the simulation model areavailable on the World Wide Web at the following URL:

http://www.nd.edu/~quake/Additionally, all control designs reported in this special issue of the Journal are avaliable at thisURL. If you cannot access the World Wide Web or have questions regarding the benchmark prob-lem, please contact the senior author via e-mail at: [email protected].

Table 1: Evaluation Criteria for the Sample Controller.

0.283 0.4560.440 0.6810.510 0.6690.513 0.7710.628 1.28

Figure 6. Loop Gain Transfer Function for Sample Control Design.

Frequency (Hz)

Mag

nitu

de (

dB)

0 10 20 30 40 50 60 70 80 90 100−50

−40

−30

−20

−10

0

10

20

30

J1 J6J2 J7J3 J8J4 J9J5 J10


ACKNOWLEDGMENTS

This research is partially supported by National Science Foundation Grant Nos. CMS 95–00301 and CMS 95–28083 (Dr. S.C. Liu, Program Director). The input provided by the Commit-tee on Structural Control, ASCE Structural Division is also acknowledged. The assistance of Dr.Erik A. Johnson in editing, evaluating the control designs, and preparing the Web site results isgratefully appreciated.

APPENDIX I – REFERENCES

Antoniou, A. (1993). Digital Filters: Analysis, Design, and Applications, McGraw-Hill, Inc., NewYork, pp. 444–446.

Chung, L.L., Lin, R.C., Soong, T.T. and Reinhorn, A.M. 1989. Experiments on Active Control forMDOF Seismic Structures,” J. of Engrg. Mech., ASCE, Vol. 115, No. 8, pp. 1609–27.

Dyke, S.J., Spencer Jr., B.F., Belknap, A.E., Ferrell, K.J., Quast, P., and Sain, M.K. (1994a). “Ab-solute Acceleration Feedback Control Strategies for the Active Mass Driver.” Proc. FirstWorld Conference on Structural Control, Pasadena, California, August 3–5, 1994, Vol. 2,pp. TP1:51–TP1:60.

Dyke, S.J., Spencer Jr., B.F., Quast, P., Sain, M.K., Kaspari Jr., D.C. and Soong, T.T. (1994b).“Experimental Verification of Acceleration Feedback Control Strategies for An ActiveTendon System,” National Center for Earthquake Engineering Research Technical ReportNCEER–94–0024, August 29.

Dyke, S.J., Spencer Jr., B.F., Quast, P., Kaspari Jr., D.C. and Sain, M.K. (1996a). “Implementationof an Active Mass Driver Using Acceleration Feedback Control.” Microcomputers in CivilEngrg., Vol. 11, pp. 305–323.

Dyke, S.J., Spencer Jr., B.F., Quast, P., Sain, M.K., Kaspari Jr., D.C. and Soong, T.T. (1996b).“Acceleration Feedback Control of MDOF Structures.” J. Engrg. Mech., ASCE, Vol. 122,No. 9, pp. 907–918.

Fujino, Y., Soong, T.T. and Spencer Jr., B.F. (1996). “Structural Control: Basic Concepts and Ap-plications.” Proceedings of the ASCE Structures Congress XIV, Chicago, Illinois, pp.1277–1287.

Housner, G.W., Masri, S.F., and Chassiakos, A.G., Eds. (1994a). Proceedings of the First WorldConference on Structural Control, International Association for Structural Control, LosAngeles.

Housner, G.W., Soong, T.T. and Masri, S. (1994b). “Second Generation of Active Structural Con-trol in Civil Engineering.” Proc. First World Conference on Structural Control, Pasadena,California, August 3–5, 1994, Vol. 1, pp. Panel:3–18.

Ivers, D.E. and Miller, L.R. (1991). "Semi-Active Suspension Technology: An EvolutionaryView." DE-Vol. 40, Advanced Automotive Technologies, (S.A. Velinsky, R.H. Fries and D.Wang, Eds.), ASME Book No. H00719, pp. 327-346.

MATLAB (1994). The Math Works, Inc. Natick, Massachusetts.Quast, P., Sain, M.K., Spencer Jr., B.F. and Dyke, S.J. (1995). “Microcomputer Implementations

of Digital Control Strategies for Structural Response Reduction,” Microcomputers in CivilEngrg., Vol. 10, pp. 13–25.

SIMULINK (1994). The Math Works, Inc. Natick, Massachusetts.


Skelton, R.E. (1988). Dynamic Systems Control: Linear Systems Analysis and Synthesis. Wiley,New York.

Soong, T.T. Active Structural Control: Theory and Practice, Longman Scientific and Technical,Essex, England, 1990.

Stengel, R.F. (1986). Stochastic Optimal Control: Theory and Application. Wiley, New York.Yao, J.T.P. (1972). “Concept of Structural Control.” ASCE J. Struct. Div., Vol. 98, pp. 1567–1574.

APPENDIX II – NOMENCLATURE

– state space matrices for the evaluation model

– state space matrices for the reduced order model

– state space matrices for the discrete controller

– interstory drift of the ith floor – feedback compensator functions – transfer function of the filters used to obtain the pseudo absolute velocities – performance function for LQG control design – ith evaluation criteria

– full state feedback gain matrix – discrete time step index – state estimator gain matrix – algrebraic Riccati matrix solution for regulator – weights in LQG control design – algrebraic Riccati matrix solution for state estimator – magnitude of the constant two-sided spectral density for the white

noises modeling the measurement noise in the LQG control design – magnitude of the constant two-sided spectral density for the white

noise used to model the ground excitation in the LQG control design – sampling time – scalar control input – scalar control input at time – measurement noise vector for the evaluation model – measurement noise vector for the control design model – state vector for the evaluation model – state vector for the discrete controller at time – state vector for the control design model – estimated state vector for the control design model – displacement of the ith floor relative to the ground – displacement of the actuator piston relative to the third floor

A B E, ,Cy Dy,Fy,

Cz,Dz,Fz

A r Br Er, ,

Cyr Dyr Fyr, ,

Ac Bc,

Cc Dc,di

f 1.( ) f 2

.( ),Hx

˜˙x s( )

JJiKkLP

Q r,S

Svivi

Sxgxg

Tu

uk t kT=v

vrx

xkc

t kT=xrxrxixm


– peak third floor displacement response relative to the ground ofthe uncontrolled building for each respective historical earthquake

– velocity of the ith floor relative to the ground – peak third floor velocity response relative to the ground of

the uncontrolled building for each respective historical earthquake – pseudo absolute velocity of the ith floor – velocity of the actuator piston relative to the third floor – pseudo absolute velocity of the actuator piston – pseudo absolute velocity of the ground – absolute acceleration of the ith floor – absolute acceleration of the actuator piston – absolute acceleration of the ground

– peak third floor absolute acceleration of the uncontrolled buildingfor each respective historical earthquake

– vector of directly measured responses – vector of directly measured responses sampled at time – output vector for the control design model – vector of responses available for calculation of the control – vector of responses available for calculation of the control sampled at time t = kT – vector of regulated responses – rms interstory drift of the ith floor – rms control signal – rms displacement of the actuator piston relative to the third floor – rms ground acceleration – worst-case stationary rms displacement of the third floor of th

uncontrolled building relative to the ground – rms velocity of the actuator piston relative to the third floor – worst-case stationary rms velocity of the third floor of the

uncontrolled building relative to the ground – worst-case stationary rms absolute acceleration of the third floor of the

uncontrolled building – rms absolute acceleration of the ith floor

, – parameters of the Kanai–Tajimi spectrum

x3o

xix3o

x˜˙aixm

x˜˙amx˜˙g

xaixamxg

xa3o

yyk t kT=yry˜y

˜ kz

σdi

σuσxm

σ xg

σx3o

σxm

σx3o

σ xa3o

σ xai

ωg ζg


et al.

BENCHMARK PROBLEMS IN STRUCTURAL CONTROL:PART II – ACTIVE TENDON SYSTEM

B.F. Spencer Jr.,

1

S.J. Dyke

2

and H.S. Deoskar

3

SUMMARY

In a companion paper (Spencer,

et al.

1997), an overview and problem definition was pre-sented for a well-defined benchmark structural control problem for a model building configuredwith an active mass driver (AMD). A second benchmark problem is posed here based on a high-fi-delity analytical model of three-story, tendon-controlled structure at the National Center forEarthquake Engineering Research (NCEER) (Chung,

et al.

1989; Dyke,

et al.

1996). The purposeof formulating this problem is to provide another setting in which to evaluate the relative effec-tiveness and implementability of various structural control algorithms. To achieve a high level ofrealism, an

evaluation

model is presented in the problem definition which is derived directly fromexperimental data obtained for the structure. This model accurately represents the behavior of thelaboratory structure and fully incorporates actuator/sensor dynamics. As in the companion paper,the evaluation model will be considered as the real structural system. In general, controllers thatare successfully implemented on the evaluation model can be expected to perform similarly in thelaboratory setting. Several evaluation criteria are given, along with the associated control designconstraints.

EXPERIMENTAL STRUCTURE

The structure on which the evaluation model is based is the actively controlled, three-story,single-bay, model building considered in Chung,

et al.

(1989). The test structure, shown in Figs. 1and 2, has a mass of 2,950 kg, distributed among the three floors, and is 254 cm in height. The ra-tio of model quantities to those corresponding to the prototype structure are: force = 1:16, mass =1:16, time = 1:2, displacement = 1:4 and acceleration = 1:1. Due to the time scaling, the naturalfrequencies of the model are approximately twice those of the prototype. The first three modes ofthe model system are at 2.27 Hz, 7.33 Hz, and 12.24 Hz, with associated damping ratios given, re-spectively, by 0.6%, 0.7%, and 0.3%.

A hydraulic control actuator, four pretensioned tendons, and a stiff steel frame connectingthe actuator to the tendons are provided to apply control forces to the test structure. The four diag-onal tendons transmit the force from the control actuator to the first floor of the structure, and thesteel frame connects the actuator to the tendons. Because hydraulic actuators are inherently open-loop unstable, a feedback control system is employed to stabilize the control actuator and improveits performance. This feedback signal is a combination of the position, velocity and pressure mea-surements. For this actuator, an LVDT (linear variable differential transformer), rigidly mountedto the piston, provides the displacement measurement, which is the primary feedback signal. Thismeasurement is also sent through an analog differentiator to provide a velocity measurement, anda pressure transducer across the actuator piston provides the pressure measurement.

1. Prof., Dept. of Civil Engrg. and Geo. Sci., Univ. of Notre Dame, Notre Dame, IN 46556-0767.2. Assist. Prof., Dept. of Civil Engrg., Washington Univ., St. Louis, MO 63130-4899.3. Former Grad. Assist., Dept. of Civil Engrg. and Geo. Sci., Univ. of Notre Dame, Notre Dame, IN 46556-0767.

Published in

Earthquake Engineering and Structural Dynamics

,

27

(11), 1998, 1141–1147.


et al.

The structure wasfully instrumented toprovide for a completerecord of the motionsundergone by the struc-ture during testing. Ac-celerometers positionedon each floor of thestructure measured theabsolute accelerations,and an accelerometer lo-cated on the base mea-sured the groundexcitation, as shown inFig. 1. The displace-ment of the actuator wasmeasured using theLVDT mentionedabove. Force transduc-ers were placed in serieswith each of the fourtendons and their indi-vidual outputs werecombined to determinethe total force applied tothe structure. Addition-al measurements werealso available for modelverification. Displace-ment transducers on thebase and on each floorwere attached to a fixedframe (

i.e.

, not attachedto the earthquake simu-lator), as shown in Fig. 1, to measure the absolute displacements of the structure and of the base.The relative displacements were determined by subtracting the base displacement from the abso-lute displacement of each floor. Because a fixed frame was necessary to measure the relative dis-placements of the structure, these measurements would not be directly available in a full-scaleimplementation. Thus, the displacement measurements are used only for model verification andare not available for feedback in the control system.

EVALUATION MODEL

A high-fidelity, linear time-invariant state space representation of the input-output model forthe structure described in the previous section has been developed. The model has 20 states and isof the form

xg

xa1

xa2

xa3

control actuator

xp

DSP Board

& Control

Computer

connecting frame

displacementtransducers

Figure 1. Schematic of Experimental Setup.

fixed frame

tendons


et al.

(1)

(2)

(3)

where is the state vector, isthe scalar ground acceleration, is the scalar control input,

isthe vector of responses that canbe directly measured,

z

= [

,

, , ,

,

, , ,

,

, , , is the vectorof responses that can be regulat-ed. Here, is the displacementof the

i

th floor relative to theground, is the displacementof the control actuator, is theabsolute acceleration of the

i

thfloor, is the vector of measure-ment noises, and

, and are matrices of appropriate

dimension. The coefficient matri-ces in Eqs. (1–3) are determinedfrom the data collected at theNCEER using the identificationmethods presented in Dyke,

et al

.(1994a,b, 1996a,b). The resultingmodel represents the input-out-put behavior of the structuralsystem up to 50 Hz and includesthe effects of actuator/sensor dy-namics and control-structure interaction. Figure 3 provides a representative comparison betweenselected model transfern functions and the experimental data.

The model given in Eqs. (1–3) is termed the

evaluation

model and will be used to assess theperformance of candidate controllers; that is, the evaluation model is considered herein to be thetrue representation of the structural system.

CONTROL DESIGN PROBLEM

The control design problem stated here follows the companion paper (Spencer,

et al.

1997)and is stated as: determine a discrete-time, feedback compensator of the form

Figure 2. Three-Degree-of-Freedom Test Structure.

x Ax Bu E xg+ +=

y Cyx Dyu Fyxg v+ + +=

z Czx Dzu Fzxg+ +=

x xgu

y xp xa1 xa2 xa3 f xg, ,,, ,[ ]′=

x1x2 x3 xP x1 x2 x3 xP xa1xa2 xa3 xaP f ]′

xi

xpxai

v

A B E Cy Dy Cz Dz, , , , , , FyFz


et al.

0 5 10 15 20 25 30 35 40 45 50−200

−100

0

100

200

0 5 10 15 20 25 30 35 40 45 50−80

−60

−40

−20

0

20

0 5 10 15 20 25 30 35 40 45 50−200

−100

0

100

200

0 5 10 15 20 25 30 35 40 45 50−60

−40

−20

0

20

Figure 3. Representative Comparison of the Transfer Functions for the Test Structure and the Evaluation Model. (a) Actuator Command to the Force in the Tendons,

(b) Actuator Command to the Third Floor Absolute Acceleration.

Frequency (Hz)

Frequency (Hz)

Pha

se (

deg)

Mag

nitu

de (

dB)

Pha

se (

deg)

Mag

nitu

de (

dB)

(a)

(b)


et al.

(4)

(5)

where , and are the state vector for the compensator, the output vector and the controlcommand, respectively, at time . For this problem, is required, and the per-formance of all control designs must be assessed using the evaluation model described previously.For each proposed control design, performance and stability robustness should be discussed. Asdetailed in the following paragraphs, the merit of a controller will be based on criteria given interms of both rms and peak response quantities. Normally, smaller values of the evaluation criteriaindicate superior performance.

Evaluation Criteria: RMS Responses

Assume that the input excitation is a stationary random process with a spectral densitydefined by the Kanai-Tajimi spectrum

(6)

where and are unknown, but assumed to lie in the following ranges:, . To have a basis for comparison, the spectral inten-

sity is chosen such that the rms value of the ground motion takes a constant value of

g

’s,

i.e.,

g

2

.sec (7)

The first criterion on which controllers will be evaluated is based on their ability to mini-mize the maximum rms interstory drift due to all admissible ground motions. Therefore, the non-dimensionalized measure of performance is given by

(8)

where is the stationary rms interstory drift for the

i

th floor, and cm is the worst-case stationary rms displacement of the third floor of the uncontrolled building over the class ofexcitations considered (occurring when rad/sec, ). The interstory drifts aregiven respectively by , and .

A second evaluation criterion is given in terms of the maximum rms absolute acceleration,yielding a performance measure given by

xk 1+c

f 1(xkc y

˜ kuk k, ), ,=

uk f 2(xkc y

˜ kk, ),=

xkc y

˜ kuk

t kT= dim(xc) 12≤

xg

Sxgxgω( )

S0(4ζg2ωg

2ω2 ωg4)+

(ω2 ωg2)–2

4ζg2ωg

2ω2+

-------------------------------------------------------=

ωg ζg8 rad/sec ωg 50 rad/sec≤ ≤ 0.3 ζg 0.75≤ ≤

σ xg3.4 10

2–⋅=

S0

2.35 103– ζg⋅

πωg(4ζg2

1)+---------------------------------=

J1 maxσd1

σx3o

---------σd2

σx3o

---------σd3

σx3o

---------, ,

=ωg ζg,

σdiσx3o

2.34=

ωg 14.5= ζg 0.3=d1 t( ) x1 t( )= d2 t( ) x2 t( ) x1 t( )–= d3 t( ) x3 t( ) x2 t( )–=


(9)

where is the stationary rms acceleration for the ith floor, and g’s is the worst-case stationary rms acceleration of the third floor of the uncontrolled building (occurring when

rad/sec, ). The hard constraints for the control effort are given by volt, kN and

. Additionally, candidate controllers are to be evaluated based on the required controlresources. Three quantities, and , should be examined to make the assessment. Therms actuator displacement, , provides a measure of the required physical size of the device.The rms actuator velocity, , provides a measure of the control power required. The rms abso-lute acceleration provides a measure of the magnitude of the forces that the actuator must gen-erate to execute the commanded control action. Therefore, the nondimensionalized controlresource evaluation criteria are

(10)

(11)

(12)

where cm/sec is the worst-case stationary rms velocity of the third floor relative tothe ground for the uncontrolled structure (occurring when rad/sec, ), and is the weight of the building (289 kN).

Evaluation Criteria: Peak Responses

Here, the input excitation is assumed to be a historical earthquake record. Both the 1940El Centro NS record and the NS record for the 1968 Hachinohe earthquake are considered. Be-cause the system under consideration is a scale model, the time scale should be increased by a fac-tor of 2 (i.e., the earthquakes occur in 1/2 the recorded time). The required scaling of themagnitude of the ground acceleration is 1. The evaluation criterion is based on minimization ofthe nondimensionalized peak interstory drifts due to both earthquake records. For each earth-quake, the maximum drifts are nondimensionalized with respect to the uncontrolled peak thirdfloor displacement, denoted , relative to the ground. Therefore, the performance measure isgiven by

(13)

J2 maxσ xa1

σ xa3o

-----------σ xa2

σ xa3o

-----------σ xa3

σ xa3o

-----------, ,

=ωg ζg,

σ xaiσ xa3o

0.485=

ωg 14.5= ζg 0.3=σu 1≤ σ f 4≤

σxp1 cm≤

σxpσxp

, σ fσxp

σxp

σ f

J3 maxσxp

σx3o

---------

=ωg ζg,

J4 maxσxp

σx3o

---------

=ωg ζg,

J5 maxσ f

W------

=ωg ζg,

σx3o33.3=

ωg 14.5= ζg 0.3= W

xg

x3o

J6 max maxd1 t( )x3o

---------------d2 t( )x3o

---------------d3 t( )x3o

---------------, ,

=tEl Centro

Hachinohe


A second performance evaluation criterion is given in terms of the peak acceleration, yield-ing

(14)

where the accelerations are nondimensionalized by the peak uncontrolled third floor acceleration,denoted , corresponding respectively to each earthquake.

The control constraints are volts, cm, kN, and both the El Centro and the Hachinohe earthquakes should again be considered. Addition-ally, the candidate controllers are to be evaluated in terms of the required control resources as fol-lows

(15)

(16)

(17)

where is the peak uncontrolled third floor relative velocity corresponding respectively to eachearthquake.

For the half-scale El Centro earthquake, cm, cm/sec and g’s. For the half-scale Hachinohe earthquake, cm, cm/sec

and g’s.

CONTROL IMPLEMENTATION CONSTRAINTS

With the following exceptions, the implementation constraints are identical to those speci-fied in the companion paper (Spencer, et al., 1997): The measurements that are directly availablefor use in determination of the control action are . The pseudo ve-locities, , are also available for determination of the control action, resulting in acombined output vector given by .

CLOSURE

The 20-state evaluation model, as well as the 10-state control design model, the MATLAB(1994) m-file used to do the sample control design, the input data and the simulation model areavailable on the World Wide Web at the following URL:

http://www.nd.edu/~quake/

J7 max maxxa1 t( )xa3o

-----------------xa2 t( )xa3o

-----------------xa3 t( )xa3o

-----------------, ,

=tEl Centro

Hachinohe

xa3omax u t( ) 3≤ t

max xp t( ) 3≤ t

max f t( ) 12≤ t

J8 max max xp t( )x3o

---------------=tEl Centro

Hachinohe

J9 max max xp t( )x3o

---------------=tEl Centro

Hachinohe

J10 max max f t( )W

-------------=tEl Centro

Hachinohe

x3o

x3o 6.45= x3o 99.9=xa3o 1.57= x3o 3.78= x3o 56.1=

xa3o 0.778=

y xp xa1 xa2 xa3 f xg, , , , ,[ ]′=x˜˙a1 x

˜˙a2 x

˜˙a3 x

˜˙g, , ,

y˜

xp xa1 xa2 xa3 f xg x˜˙a1 x

˜˙a2 x

˜˙a3 x

˜˙g, , , , , , , , ,[ ]′=


Additionally, all control designs reported in this special issue of the Journal are avaliable at thisURL. If you cannot access the World Wide Web or have questions regarding the benchmark prob-lem, please contact the senior author via e-mail at: [email protected].

ACKNOWLEDGMENTS

This research is partially supported by National Science Foundation Grant Nos. CMS 95–00301 and CMS 95–28083 (Dr. S.C. Liu, Program Director). The input provided by the Commit-tee on Structural Control, ASCE Structural Division is also acknowledged. The assistance of Dr.Erik A. Johnson in editing, evaluating the control designs, and preparing the Web site results isgratefully appreciated.

APPENDIX I – REFERENCES

Chung, L.L., Lin, R.C., Soong, T.T. and Reinhorn, A.M. 1989. “Experiments on Active Controlfor MDOF Seismic Structures,” J. of Engrg. Mech., ASCE, Vol. 115, No. 8, pp. 1609–27.

Dyke, S.J., Spencer Jr., B.F., Belknap, A.E., Ferrell, K.J., Quast, P., and Sain, M.K. (1994a). “Ab-solute Acceleration Feedback Control Strategies for the Active Mass Driver.” Proc. FirstWorld Conference on Structural Control, Pasadena, California, August 3–5, 1994, Vol. 2,pp. TP1:51–TP1:60.

Dyke, S.J., Spencer Jr., B.F., Quast, P., Sain, M.K., Kaspari Jr., D.C. and Soong, T.T. (1994b).“Experimental Verification of Acceleration Feedback Control Strategies for An ActiveTendon System,” National Center for Earthquake Engineering Research Technical ReportNCEER–94–0024, August 29.

Dyke, S.J., Spencer Jr., B.F., Quast, P., Kaspari Jr., D.C. and Sain, M.K. (1996a). “Implementationof an Active Mass Driver Using Acceleration Feedback Control.” Microcomputers in CivilEngrg., Vol. 11, pp. 305–323.

Dyke, S.J., Spencer Jr., B.F., Quast, P., Sain, M.K., Kaspari Jr., D.C. and Soong, T.T. (1996b).“Acceleration Feedback Control of MDOF Structures.” J. Engrg. Mech., ASCE, Vol. 122,No. 9, pp. 907–918.

MATLAB (1994). The Math Works, Inc. Natick, Massachusetts.Spencer Jr., B.F., Dyke, S. and Deoskar, H. (1997). “Benchmark Problems in Structural Control:

Part I – Active Mass Driver System.” Earthquake Engrg. and Struct. Dyn., to appear.

APPENDIX II – NOMENCLATURE (SUPPLEMENT TO COMPANION PAPER)

– displacement of the actuator piston relative to the ground – velocity of the actuator piston relative to the ground – net force in the tendons

– rms displacement of the actuator piston relative to the ground – rms velocity of the actuator piston relative to the ground – rms force in the tendons

xpxp

fσxp

σxp

σ f

Documents

BENCHMARK PROBLEMS IN STRUCTURAL …sstl.cee.illinois.edu/benchmarks/bench1journal/BENCH.pdf · relative effectiveness and implementability of various structural control algorithms